On the chromatic number of random d-regular graphs

• Published in: Advances in mathematics (New York. 1965) Vol. 223; no. 1; pp. 300 - 328
• Main Authors: Kemkes, Graeme, Pérez-Giménez, Xavier, Wormald, Nicholas
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2010
• Subjects: Random graphs; Random graphs
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2009.08.006

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d < 2 ( k − 1 ) log ( k − 1 ) . From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k − 1 or k. If moreover d > ( 2 k − 3 ) log ( k − 1 ) , then the value k − 1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k + 1 . Our proof applies the small subgraph conditioning method to the number of equitable k-colourings, where a colouring is equitable if the number of vertices of each colour is equal.